Steinberg fixed-point group

Let $\mathbf G$ be a connected simply connected semisimple algebraic group over $\overline{\mathbb F}_p$ . For an algebraic-group endomorphism $F$ such that $F^a=F_{p^b}$ for some positive integers $a,b$ , the Steinberg fixed-point group and its central quotient are

\mathbf G^F=\{g\in\mathbf G(\overline{\mathbb F}_p):F(g)=g\}, \qquad \mathbf G^F/Z(\mathbf G^F).

Here $F_{p^b}$ is the Frobenius endomorphism of a chosen split model and $Z$ is the finite center. The fixed points form a finite group under inherited multiplication. A specific twisted family includes the choice of $\mathbf G$ and $F$ as part of its definition.

Graph and exceptional twists

For a pinned Dynkin-diagram automorphism $\gamma$ of order $a$ commuting with $F_q$ , the choice $F=\gamma F_q$ has $F^a=F_{q^a}$ . An exceptional endomorphism $\tau$ in the Suzuki–Ree cases instead satisfies $\tau^2=F_p$ . With $F=\tau F_{p^m}$ , one has $F^2=F_{p^{2m+1}}$ .

These two constructions use different field conventions. Here ${}^2A_{n-1}(q)$ uses matrices over $\mathbb F_{q^2}$ , while ${}^3D_4(q)$ and ${}^2E_6(q)$ use the indicated graph–Frobenius parameter $q$ . Some sources put $q^2$ or $q^3$ in those names instead.

References

Robert Steinberg, Lectures on Chevalley Groups (Yale, 1967), §11, especially Theorems 34–35, PDF pp. 193–196; notation for field and twisting parameters differs.
Robert A. Wilson, On the simple groups of Suzuki and Ree (24 April 2010), Introduction; §5.1 orders, §5.2 simplicity, and §5.3 the groups over the prime fields, pp. 24–30.